Physics-informed Neural Operators for Predicting 3D Electromagnetic Fields Transformed by Metasurfaces

TL;DR

This work tackles the computational bottleneck of predicting 3D electromagnetic fields transformed by complex metasurfaces. It introduces physics-informed Fourier neural operators that map 3D permittivity distributions to EM fields while enforcing Maxwell's equations via Maxwell residuals, trained on a large FDTD-generated database of synthetic metasurfaces. The approach achieves high accuracy across diverse geometries, including unseen designs, with up to a 67× speedup over full-wave solvers and an intrinsic capability for super-resolution. These results enable rapid, gradient-based inverse-design workflows for nanostructured EM materials and demonstrate a data-efficient, resolution-flexible surrogate modeling paradigm for 3D metasurface simulations.

Abstract

Metasurfaces, typically realized as arrays of nanopillars, transform electromagnetic (EM) fields depending on their geometry and spatial arrangement. For solving the inverse problem of designing new metasurfaces that transform EM fields in a desirable manner, it is often necessary to explore large design spaces through full-wave simulations that can be computationally demanding. In this work, we demonstrate that neural operators, which are artificial neural network architectures designed to learn operators between function spaces, can effectively approximate the differential operators underlying Maxwell's equations, enabling their use as fast and accurate 3D surrogate models that can predict 3D EM fields transformed by metasurfaces. To calibrate neural operators, we generate synthetic training data consisting of 3D metasurface geometries together with their associated 3D EM fields obtained by numerically solving Maxwell's equations. Using the generated synthetic data, we train physics-informed neural operators to minimize physical inconsistencies of predicted EM fields by incorporating residuals that capture deviations from Maxwell's equations. We observe that a training dataset consisting of fewer than 5000 examples already suffices to achieve reasonable results. In particular, our experiments show that the resulting 3D surrogate model achieves high predictive performance across a wide range of metasurface geometries, including types of structures not encountered during training. Notably, it predicts diffraction efficiencies with relative errors of 3.9 % and provides a 67-fold speedup compared to conventional 3D simulations. Overall, once trained, our 3D surrogate model can rapidly predict EM fields for previously unseen metasurface geometries, which can facilitate efficient gradient-based design of nanostructured materials for EM wave control.

Figures (15)

• Figure 1: Generated metasurfaces. Cross-sections parallel to the $x$-$y$-plane of generated metasurfaces associated with the (a) disks-only, (b) squares-only, (c) disks-squares, (d) freeform-only and (e) freeform-disks-squares scenarios. Perpendicular cross-sections of (e) for the planes with $y=3200nm$ and $x=3200nm$ are visualized in (f) and (g), respectively.
• Figure 2: Training data. Cross-section of a 3D distribution $\varepsilon_\mathrm{r}^\mathrm{3D}$ of relative permittivities (a) and associated magnitude of the simulated magnetic field $\mathbf{H}_\mathrm{d}$ (b).
• Figure 3: Network architecture. Input tensors are single-channel 3D distributions $\varepsilon_\mathrm{r}^\mathrm{3D}$ that are processed through five Fourier layers, the first four of which are deploying a GeLU activation function, whereas the last Fourier layer is followed by a scaled tanh activation function. The labels above Fourier layers indicate the truncation limit (m) and the number of output channels (c). For example, the label m32c64 indicates a Fourier layer with truncation limits $m_\mathrm{x}=m_\mathrm{y}=m_\mathrm{z}=32$ and $c=64$ output channels. For predicted fields ${\mathbf{H}}_\mathrm{d}^\mathrm{pred}$ the data and Maxwell losses loss $L_\mathrm{data}$, $L_\mathrm{Maxwell}$. The total loss $L$ is defined as a weighted sum of these contributions.
• Figure 4: Loss evolution. Training (a) and validation (b) loss $L$ for FNO, FNO-SD, 3D-WaveY-Net and 3D-WaveY-Net-SD. Validation loss values for $L_\mathrm{data}, L_\mathrm{Maxwell}$ and $L$ during the training of FNO are shown in (c).
• Figure 5: Visual validation. Ground truth of the absolute values of ${\mathbf{H}}_\mathrm{d}$ in a planar section (a), and the corresponding absolute values of ${\mathbf{H}}_\mathrm{d}^\mathrm{pred}$predicted by FNO (b), FNO-L2 (c), FNO-SD (d), 3D-WaveY-Net (e) and 3D-WaveY-Net-SD (f).